3. Where do we begin?

The first question popping up is where we might begin
looking for this hypothetical language of Quantum Physics.
One is naturally lead to think that a departing point is
the Bohr's correspondence principle, who
originated the quantum theory by trying to reveal its
mathematical substrate.

The Bohr's correspondence principle—accordingly to which
Classical Physics has to be treated as a limit case of the
Quantum one—was the first heuristic method leading to the
construction of mathematical models for quantum phenomena.
Taking into account that the natural language for Classical
Physics is Differential Calculus, it may be assumed that
Differential Calculus is a limit case—or, better to say, a
degenerate case—of a more general mathematical theory going
under the name of "Quantistic" Differential Calculus. This
general idea represents the mathematical paraphrase of the
correspondence principle, the mathematical correspondence
principle. This principle lead us to think that the
mathematical language of Quantum Physics has to be a
natural extension of the Differential Calculus,
instead of a chaos of appendages and superstructures formed
by many constructions of Functional Analysis, Measure
Theory, Non-commutative Algebra, and so on.
The mathematical correspondence principle is not just a
mere indication of the existence of a mysterious Quantistic
Differential Calculus. Indeed it can be notice that the
ordinary differential equations of Classical Mechanics
describe the behavior of the singularities of the solutions
of the differential equations of the Quantum Mechanics.
That are equations in partial derivatives. On the other
hand, the departing point for the theory of quantum fields
is the equations of classical fields, that by analogy
describe the behavior of the singularities of the quantum
fields. But classical fields are described by means of
partial differential equations, so that—assuming the
validity of the correspondence principle in this situation
as well—we are brought to the conclusion that the
equations describing quantum fields undoubtedly have to be
differential equations of an unknown first genre being in
the same relationship with partial differential equations
as the partial differential equations are with the ordinary
ones. This way the mathematical correspondence
principle not only tells about the existence of the Quantum
Differential Calculus, but also states that it must be
looked for within a duly developed theory of partial
differential equations.
Historically Bohr's principle has been used only
for technical purposes, allowing to guess some
quantization schemes. Its hidden mathematical aspect can be
enlightened and comprehended in the era of formation of the
existing quantum theory. One reason for this was the
marginal state of the studies of the non linear PDEs, while
everybody believed—accordingly to current trends—that truth
was hidden into Hilbert spaces. Therefore von Neumann's
statement that theories of self-adjoint operators on
Hilbert spaces would represent the mathematical foundations
of Quantum Mechanics was enthusiastically embraced by the
wide public in spite of the fact that it violated Bohr's
correspondence principle in a shocking rough way. It has
been rumored that only Levi-Civita tried very
timidly to display his own disagreement, but his
subdued voice was not heeded. It was an "ecological
catastrophe", whose consequences would live on for a long
time.